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Descriptive statistics — means, standard deviations, graphs — summarise data, but they cannot tell us whether a difference or relationship is real or merely a fluke of sampling. Suppose one group recalls a mean of 14 words and another 12: is that a genuine effect, or could it easily arise by chance? Inferential statistics answer exactly this question. They allow psychologists to infer something about the wider population from a sample by calculating the probability that the results occurred by chance, and on that basis deciding whether to reject the null hypothesis. Without this step a researcher could only ever describe their particular sample; with it, they can make justified claims about people in general — which is what turns a one-off observation into scientific knowledge. This lesson covers the logic of the whole enterprise: probability and the 0.05 significance level, the null and alternative hypotheses, one- and two-tailed testing, the comparison of the calculated (observed) value with the critical value, and the two ways a statistical decision can go wrong — Type I and Type II errors. The choice of which specific test to run is the subject of the next lesson; here we build the reasoning that every test relies on.
Key Definition: Inferential statistics are statistical tests used to decide whether the results of a study are statistically significant — unlikely to have occurred by chance — so that a conclusion can be generalised from the sample to the wider population.
By the end of this lesson you will be able to:
Edexcel 9PS0 — Paper 3: Psychological Skills (Research Methods). This lesson develops the significance-testing logic assessed in Section A of Paper 3, where candidates must interpret significance, state and use hypotheses, and reason about error — all in the context of a described study. Our sequence leads with the logic of probability before the mechanics of comparing values, so the ordering reflects our teaching rationale rather than the specification's own.
| Our lesson covers | Edexcel 9PS0 research-methods area |
|---|---|
| Probability and the p≤0.05 significance level | Probability and significance |
| Null and alternative hypotheses | Hypotheses in statistical testing |
| One-tailed vs two-tailed tests | Directional / non-directional testing |
| Critical value vs observed/calculated value | Use of critical values in interpreting significance |
| Type I and Type II errors | Errors in statistical decision-making |
Assessment Objectives. These items are strongly AO2 (applying significance and error reasoning to a novel study; stating the correct hypothesis; comparing calculated and critical values) and AO3 (evaluating a significance decision and the consequences of an error), on an AO1 base of accurate definitions. Questions may require you to read a critical-values table, state a conclusion in context, or reason about which error is more serious in a given scenario.
Connects to…
In psychology, the conventional significance level is
p≤0.05
meaning that the probability the observed results are due to chance is 5% or less — equivalently, we can be at least 95% confident the effect is real. If a test shows the probability of a chance result is at or below this threshold, we reject the null hypothesis and accept the alternative.
| Significance level | Interpretation |
|---|---|
| p ≤ 0.05 | The standard level — results judged statistically significant |
| p ≤ 0.01 | More stringent (1% chance of a false positive) — used when a false positive would be serious, e.g. trialling a drug |
| p ≤ 0.10 | More lenient — occasionally used in exploratory or pilot research |
Why 5%? The figure is a convention, not a law of nature. It represents a pragmatic balance: strict enough that we will not constantly cry "effect!" over random noise (which a 10% level would risk), but lenient enough that genuine effects of reasonable size can be detected (which a 1% level might miss). Probability itself runs on a scale from 0 (an event is impossible) to 1 (it is certain), so p≤0.05 simply marks the point at which a chance explanation becomes implausible enough to discard. Crucially, "significant" never means "certain": there remains up to a 5% chance that we have rejected a true null hypothesis. Replication is therefore essential — a single significant result could always be the 1-in-20 fluke.
Key Definition: The significance level is the probability threshold below which the null hypothesis is rejected. In psychology, p≤0.05 is standard: there is a 5% or smaller probability that the results occurred by chance.
Exam Tip: Note the careful wording. p≤0.05 does not mean "95% certain the hypothesis is true". It means that if the null hypothesis were true, results this extreme would occur 5% of the time or less. We never prove a hypothesis — we reject or retain the null.
Every inferential test is a contest between two statements.
Key Definition: The null hypothesis (H0) predicts no significant difference (or, in a correlation, no significant relationship): any pattern observed in the sample is due to chance. The alternative (experimental) hypothesis (H1) predicts that there is a significant difference or relationship.
The logic can seem back-to-front, because it is the null that we actually test — not the alternative we are usually hoping to support. The reason is sound. We can never prove an alternative hypothesis true; there might always be an unseen exception. But we can gather enough evidence to make the "no effect" explanation implausible. So we provisionally assume the null (no effect) and ask: if that were true, how likely is the result we obtained? If that probability is at or below the significance level, the null becomes implausible and we reject it, accepting the alternative by elimination. If the probability is higher than 0.05, we retain (fail to reject) the null — noting that this is not the same as proving there is no effect, merely that we have insufficient evidence to claim one.
The alternative hypothesis comes in two forms, and which one is stated before data collection determines whether the test is one- or two-tailed:
For a correlation the language shifts from difference to relationship: a directional correlational hypothesis predicts a positive or negative relationship, while the null predicts "no significant correlation… any correlation is due to chance." Matching the wording to the design is a frequent discriminator between a secure and a careless answer.
Worked example — writing both hypotheses. Suppose a study investigates whether caffeine speeds reaction time, using a repeated-measures design with reaction time measured in milliseconds.
Notice that every version is operationalised — the IV (200 mg caffeine vs placebo) and the DV (mean reaction time in ms on a stated task) are precisely defined — and that only the directional version contains a direction word. A hypothesis that is vague ("caffeine affects reaction time") or that omits the "due to chance" clause from the null loses marks even when the underlying reasoning is sound.
Whether a test is one- or two-tailed is fixed by the form of the hypothesis, and it changes how demanding the significance criterion is.
graph TD
A[Does the hypothesis<br/>state a direction?] -->|Yes: more / fewer / positive| B[Directional hypothesis<br/>ONE-tailed test]
A -->|No: just 'a difference'| C[Non-directional hypothesis<br/>TWO-tailed test]
B --> B1[The 5% rejection region<br/>sits in ONE tail<br/>→ critical value easier to reach]
C --> C1[The 5% is split 2.5% + 2.5%<br/>across BOTH tails<br/>→ more extreme result needed]
Exam Tip: Never choose "one-tailed" after seeing the results in order to scrape significance — that is a form of cheating that inflates Type I errors. The tail is fixed by the hypothesis before data are collected.
When a test is run, it produces a calculated (observed) value — a single number summarising the size of the effect in the data. To judge significance, this is compared against a critical value read from a statistical table.
Key Definition: A critical value is the value a test statistic must reach — either exceed, or fall at or below, depending on the test — for the result to be declared significant at the chosen level. It is looked up in a statistical table.
The table is entered using three pieces of information:
Two patterns are worth memorising. A stricter significance level (moving from 0.05 to 0.01) makes the critical value harder to satisfy, reflecting the higher bar for declaring significance. And a larger sample generally makes significance easier to reach, because a bigger N gives the test more power to distinguish a real effect from chance — one reason under-powered (small-N) studies so often fail to reach significance even when an effect exists.
The direction of the comparison — whether the calculated value must be greater than or less than or equal to the critical value — depends on the specific test, and is developed fully in the next lesson. Whichever applies, if the criterion is met the result is significant, the null hypothesis is rejected, and we conclude there is a significant difference or correlation; if it is not met, the null is retained.
Key Definition: A result is statistically significant when the calculated value satisfies the critical-value criterion at the chosen level, so that the probability of the result arising by chance is judged small enough (≤ 5%) to reject the null hypothesis.
The sign test is the simplest inferential test and a clean way to see the calculated-versus-critical logic in action. It is used when the hypothesis predicts a difference, the design is related (repeated measures or matched pairs), and the data reduce to the direction of change for each participant. Its procedure is: record each participant's score in the two conditions; find the sign of each difference (+, − or 0); discard any zeros; let N be the number of remaining participants; let S (the calculated value) be the count of the less frequent sign; then look up the critical value of S for that N, tail and significance level. If S≤ the critical value, the result is significant.
Worked example. Ten participants rate their stress before and after a relaxation technique; the hypothesis is directional (it will reduce stress).
| Participant | Before | After | Difference | Sign |
|---|---|---|---|---|
| 1 | 8 | 5 | −3 | − |
| 2 | 7 | 6 | −1 | − |
| 3 | 6 | 6 | 0 | (excluded) |
| 4 | 9 | 4 | −5 | − |
| 5 | 5 | 3 | −2 | − |
| 6 | 8 | 7 | −1 | − |
| 7 | 6 | 5 | −1 | − |
| 8 | 7 | 8 | +1 | + |
| 9 | 9 | 6 | −3 | − |
| 10 | 8 | 5 | −3 | − |
Participant 3 is excluded (difference =0), so N=9. There are 8 minus signs and 1 plus sign, so the calculated value S=1. The critical value for a one-tailed test at p≤0.05 with N=9 is 1. Since the calculated value (1)≤ the critical value (1), the result is significant: the relaxation technique significantly reduced stress ratings (p≤0.05, one-tailed), and the null hypothesis is rejected. Note that a one-tailed critical value was used because the hypothesis was directional; had the prediction merely been that stress would change, the two-tailed critical value would apply and significance would be slightly harder to reach — a concrete illustration of everything above.
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